// Numbas version: exam_results_page_options {"name": "Partial fraction breakdown A,B & C: repeated linear factor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b1": {"templateType": "randrange", "definition": "random(6..10#1)", "name": "b1", "group": "Ungrouped variables", "description": ""}, "a1": {"templateType": "randrange", "definition": "random(1..5#1)", "name": "a1", "group": "Ungrouped variables", "description": ""}, "F": {"templateType": "randrange", "definition": "random(1..6#1)", "name": "F", "group": "Ungrouped variables", "description": ""}, "H": {"templateType": "randrange", "definition": "random(2..12#1)", "name": "H", "group": "Ungrouped variables", "description": ""}}, "variable_groups": [], "functions": {}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "advice": "

\$$Q(s)=\\frac{\\var{F}s^2+\\var{H}}{(S+\\var{a1})(s+\\var{b1})^2}=\\frac{A}{s+\\var{a1}}+\\frac{B}{s+\\var{b1}}+\\frac{C}{(s+\\var{b1})^2}\$$

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Multiply across by \$$(s+\\var{a1})(s+\\var{b1})^2\$$

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\$$\\var{F}s^2+\\var{H}=A(s+\\var{b1})^2+B(s+\\var{a1})(s+\\var{b1})+C(s+\\var{a1})\$$

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let \$$s=-\\var{a1}\$$

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\$$\\var{F}(-\\var{a1})^2+\\var{H}=A(\\simplify{{b1}-{a1}})^2+B(0)+C(0)\$$

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\$$\\simplify{{F}*{a1}^2+{H}}=\\simplify{({b1}-{a1})^2}A\$$

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\$$A=\\frac{\\simplify{{F}*{a1}^2+{H}}}{\\simplify{({b1}-{a1})^2}}\$$

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let \$$s=-\\var{b1}\$$

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\$$\\var{F}(-\\var{b1})^2+\\var{H}=A(0)+B(0)+C(\\simplify{-{b1}+{a1}})\$$

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\$$\\simplify{{F}*{b1}^2+{H}}=\\simplify{(-{b1}+{a1})}C\$$

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\$$C=\\frac{\\simplify{{F}*{b1}^2+{H}}}{\\simplify{(-{b1}+{a1})}}\$$

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Compare coefficients of \$$s^2\$$

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\$$\\var{F}=A+B\$$

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\$$\\var{F}=\\frac{\\simplify{{F}*{a1}^2+{H}}}{\\simplify{({b1}-{a1})^2}}+B\$$

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\$$B=\\simplify{(({F}*({b1}^2-2*{b1}*{a1})-{H})/({b1}-{a1})^2)}\$$

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Find the partial fraction breakdown of the compound fraction:

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\$$Q(s)=\\frac{\\var{F}s^2+\\var{H}}{(s+\\var{a1})(s+\\var{b1})^2}\$$

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If the partial fraction breakdown is given by:

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\$$Q(s) =\\frac{A}{s+\\var{a1}}+\\frac{B}{s+\\var{b1}}+\\frac{C}{(s+\\var{b1})^2}\$$

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Calculate the values of \$$A, B\$$ and \$$C\$$ and give your answers as fractions

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\$$A=\$$ [[0]]

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\$$B=\$$ [[1]]

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\$$C=\$$ [[2]]

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